GARCH(n,alpha,beta,stat)

function [X,s2] = GARCH(n,alpha,beta,stat)

% alpha = (alpha_0,alpha_1,...,alpha_p)
% beta = (beta_1,...,beta_q)

euler = 0.57721566490153286061;
log2 = 0.69314718056;

if nargin<4, stat=2000; end

alpha=alpha(:);
beta=beta(:);

p = length(alpha); 
q = length(beta);

Z = randn(1,stat+n);

% valeur initiale de 1  p pour X
% valeur initiale de 1  q pour s2
X = ones(1,p+stat+n);
s2 = ones(1,q+stat+n);

for i=max([p q])+1:stat+n
    s2(i) = X(i:-1:i-p+1).^2*alpha + s2(i-1:-1:i-q)*beta;
    % s2(i) = 1+0.7*X(i-1)^2+ 0.2*s2(i-1);
    X(i) = sqrt(s2(i))*Z(i);
end

X = X(end-n+1:end);
s2 = s2(end-n+1:end);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% This is part of the package EstimHidden devoted to the estimation of 
%
% 1/ the density of X in a convolution model where Z=X+noise1 is observed 
%
% 2/ the functions b (drift) and s^2 (volatility) in an "errors in variables" 
%    model where Z and Y are observed and assumed to follow:
%           Z=X+noise1 and Y=b(X)+s(X)*noise2.
%
% 3/ the functions b (drift) and s^2 (volatility) in an stochastic
%    volatility model where Z is observed and follows:
%           Z=X+noise1 and X_{i+1} = b(X_i) + s(X_i)*noise2
%
% in any cases the density of noise1 is known. We consider three cases for
% this density : Gaussian ('normal'), Laplace ('symexp') and log(Chi2)
% ('logchi2)
%
% See function DeconvEstimate.m and examples in files ExampleDensity.m and
% ExampleRegression.m
%
% Authors : F. COMTE and Y. ROZENHOLC 
%
%
% For more information, see the following references:
%
% DENSITY DECONVOLUTION
%%%%%%%%%%%%%%%%%%%%%%%
%
% 1/ "Penalized contrast estimator for density deconvolution", 
%    The Canadian Journal of Statistics, 34, 431-452, 2006.
%    by F. COMTE, Y. ROZENHOLC, and M.-L. TAUPIN
%
% 2/ "Finite sample  penalization in adaptive density deconvolution", 
%    Journal of Statistical Computation and Simulation. 
%    Available online.
%    by F. COMTE, Y. ROZENHOLC, and M.-L. TAUPIN
%
% 3/ "Adaptive density estimation for general ARCH models", 
%    Preprint HAL-CNRS : hal-00101417  at http://hal.archives-ouvertes.fr/
%    by F. COMTE, J. DEDECKER, and  M.-L. TAUPIN. 
%
% REGRESSION and AUTO-REGRESSION
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% 4/ "Nonparametric estimation of the regression function in an
%    errors-in-variables model", 
%    Statistica Sinica, 17, n3, 1065-1090, 2007. 
%    by F. COMTE and M.-L. TAUPIN
%
% 5/ "Adaptive estimation of the dynamics of a discrete time stochastic
%    volatility model", 
%    Preprint HAL-CNRS : hal-00170740 at http://hal.archives-ouvertes.fr/
%    by F. COMTE, C. LACOUR, and Y. ROZENHOLC. 
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% You can use this software for NON-COMMERCIAL USE ONLY. 
%
% You can distribute this sofware unchanged and only unchanged, which implies
% including all files found in the folder cointainning this file.
%
% This software, and any part of it, is proposed for NON-COMMERCIAL USE 
% ONLY. 
%
% Please, contact the author for and before any non-academic use
% of this software.
%
% To reproduce this code or any part of this code in the original language 
% or in any other language, for commercial use, please contact the Author
%
% For academic purpose, cite this package and the connected papers.
%
% Corresponding author : Y. Rozenholc, [e-mail]
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Examples in files ExampleDensity.m and ExampleRegression.m
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%